Interpolation in Jacobi-weighted spaces and its application to a posteriori error estimations of the p-version of the finite element method

TL;DR

This paper develops new interpolation techniques in Jacobi-weighted spaces and applies them to create an effective residual-based a posteriori error estimator for the p-version finite element method, ensuring reliable error assessment.

Contribution

Introduces local and global interpolators in Jacobi-weighted spaces with optimal approximation order for the p-version finite element method and establishes estimator-error equivalence.

Findings

Optimal order of approximation achieved in Jacobi-weighted spaces.

Proposed a residual-type a posteriori error estimator.

Proved estimator-error equivalence in weighted norms.

Abstract

The goal of this work is to introduce a local and a global interpolator in Jacobi-weighted spaces, with optimal order of approximation in the context of the $p$-version of finite element methods. Then, an a posteriori error indicator of the residual type is proposed for a model problem in two dimensions and, in the mathematical framework of the Jacobi-weighted spaces, the equivalence between the estimator and the error is obtained on appropriate weighted norm.